Robust Mechanism Design and Implementation: A Selective Survey

Transcription

1 Robust Mechanism Design and Implementation: A Selective Survey Dirk Bergemann and Stephen Morris January 2009 UCI Conference on Adaptive Systems and Mechanism Design

2 Introduction mechanism design and implementation literatures are theoretical successes mechanisms seem too complicated to use in practise... successful applications of commonly include ad hoc restrictions simplicity, non-parametric, detail free, ex post equilibrium...

3 Weaken Informational Assumptions if the optimal solution to the planner s problem is too complicated or sensitive to be used in practice, presumably the original description of the planner s problem was itself awed can improved modelling of the planner s problem endogenously generate the robust features of mechanisms that researchers have been tempted to assume? weaken informational requirements speci cally weaken common knowledge assumption in the description of the planner s problem: the Wilson doctrine

4 The Wilson Doctrine Game theory has a great advantage in explicitly analyzing the consequences of trading rules that presumably are really common knowledge; it is de cient to the extent that it assumes other features to be common knowledge, such as one agent s probability assessment about another s preferences or information. I foresee the progress of game theory as depending on successive reductions in the base of common knowledge required to conduct useful analyses of practical problems. Only by repeated weakening of common knowledge assumptions will the theory approximate reality. Wilson (1987)

5 Weakening Common Knowledge in game theory, Harsanyi (1967/68), Mertens and Zamir (1985) established that relaxing common knowledge assumptions is equivalent to adding types... environments with incomplete information can be modeled as a Bayesian game where wlog there is common knowledge among players of (i) each player s type spaces and (ii) each type s beliefs over types of other players economic analysis assumes smaller type spaces than universal type space yet maintains common knowledge of (i) and (ii) are the implicit common knowledge assumptions that come from working with small types spaces problematic? perhaps especially in mechanism design (Neeman (2004))?

6 Our Agenda (circa 2000) introduce rich (higher order belief) types and strategic uncertainty into mechanism design literature relax (implicit) common knowledge assumptions by going from "naive" type space to "universal" type space nd robust mechanism with large type space and obtain comparative statics results across type spaces in particular 1 brie y establish a few easy benchmark "abstract" results 2 develop a close link between this robust approach and applications a decade and seven papers/notes later, we are kind of done with (1)

7 Seven Papers: A Selective Survey since 2000, Stephen Morris and I have written a series of papers on "Robust Mechanism Design": 1 "Robust Mechanism Design", Econometrica (2005) 2 "Ex Post Implementation" Games and Economic Behavior (2008) 3 "Robust Implementation in Direct Mechanisms" Review of Economic Studies (2009) 4 "An Ascending Auction for Interdependent Values" American Economic Review (2007) 5 "The Role of the Common Prior Assumption in Robust Implementation" Journal of European Economic Association (2008) 6 "Robust Implementation in General Mechanisms" (2009) 7 "Robust Virtual Implementation" Theoretical Economics (2009)

8 Payo Environment agent i 2 I = f1, 2,..., I g i s "payo type" θ i 2 Θ i payo type pro le θ 2 Θ = Θ 1 Θ I social outcome a 2 A utility function u i : A Θ! R social choice function f : Θ! A

9 Type Spaces richer type space T i than payo type space Θ i i s type is t i 2 T i, t i includes description of: payo type: bθ i : T i! Θ i bθ i (t i ) is i s payo type of t i beliefs about types T i of other players: bπ i : T i! (T i ) bπ i (t i ) is i s belief type of t i type space is a collection T = ft i, bθ i, bπ i g I i=1 type t i contains information about preferences and information of others agents, i.e. beliefs and higher-order beliefs

10 Allocating a Single Object I agents agent i has a payo type θ i 2 Θ i = [0, 1] agent i s valuation of the "object" is v i (θ 1,..., θ I ) interdependent value model (Maskin (1992), Dasgupta and Maskin (1999)) all agents have quasi-linear utility don t know anything about agent i s beliefs and higher order beliefs about θ i

11 Private Values value of i does only depend on payo type of agent i: v i (θ) = θ i second price sealed bid auction, direct mechanism i bids b i 2 [0, 1], rule of second price auction: highest bid wins, pays second highest bid truthful reporting leads to e cient allocation of object: q (θ) (e cient correspondence) ( qi 1 (θ) = #fj:θ j θ k for all kg, if θ i θ k for all k 0, if otherwise dominant strategy to truthfully report type

12 Interdependent Values linear example: v i (θ) = θ i + γ θ j j6=i generalized VCG mechanism", direct mechanism agent bids b i 2 [0, 1], highest bid wins, pays the second highest bid PLUS γ times the bid of others: max j6=i fb j g + γ b j j6=i truthful reporting is an ex post equilibrium of the direct mechanism if γ 1, cf. Maskin (1992)

13 Detour for De nitions "ex post equilibrium": each type of each agent has an incentive to tell truth if he expects all other agents to tell the truth under private values, ex post equilibrium is equivalent to dominant strategies equilibrium if truthtelling is an ex post equilibrium of the direct mechanism for an allocation rule (including transfers), then the allocation rule "ex post incentive compatible" [EPIC]

14 Incentive Compatibility De nition A scf f is interim incentive compatible on type space T if Z u i Z f (t), bθ (t) d bπ i (t i j t i ) u i f ti 0, t i, bθ (t) d bπ i (t i j t i ) t i t i for all i, t 2 T and t 0 i 2 T i. De nition A scf f is ex post incentive compatible if, for all i, θ 2 Θ, θ 0 i 2 Θ i : u i (f (θ), θ) u i f θ 0 i, θ i, θ. Compare: A scf is dominant strategy incentive compatible if for all i and all θ, θ 0 : u i f θ i, θ 0 i, θ ui f θi 0, θ 0 i, θ

15 Robust Mechanism Design I when does there exist a mechanism with the property that for any beliefs and higher order beliefs that the agents may have, there exists an equilibrium where an acceptable outcome is chosen? in single good example, consider e cient correspondence q of object and any suitable transfers

16 Robust Mechanism Design II a su cient condition is that there exists an allocation rule as a function of agents payo type that is ex post incentive compatible, i.e., in a payo type direct mechanism, each agent has an incentive to announce his type truthfully whatever his beliefs about others payo types the larger the type space, the more incentive constraints there are, the harder it becomes to implement scc from smallest type space: naive type space to largest type space: universal type space

17 Ex Post Equivalence Theorem (2005) f is ex post incentive compatible if and only if f is interim incentive compatible on every type space T. ex post equivalence can be generalized to social choice correspondence with product structure ex post equivalence fails to hold for scc in general, e.g. e cient allocation with budget balance ex post equilibrium notion incorporates concern for robustness to higher-order beliefs in private values case, ex post implementation is equivalent to dominant strategies implementation: c.f. Ledyard (1978) and Dasgupta, Hammond and Maskin (1979) in private value environments and dominant strategies

18 Ex Post Implementation when does there exist a mechanism such that, not only is there an ex post equilibrium delivering the right outcome, but every ex post equilibrium delivers the right outcome? thus there is full implementation under the solution concept of ex post equilibrium - and we call this ex post implementation in addition to ex post incentive compatibility - an ex post monotonicity condition is necessary and almost su cient ex post monotonicity condition neither implies nor is implied by Maskin monotonicity (necessary and almost su cient for implementation under complete information) generalized VCG satis es ex post monotonicity condition if I 3 and γ 6= 0

19 Robust Implementation when does there exist a mechanism with the property that for any beliefs and higher order beliefs that the agents may have, every interim equilibrium has the property that an acceptable outcome is chosen? we call this "robust implementation" this is not the same as the ex post implementation: to rule out bad equilibria, it was enough to make sure you could not construct a "bad" ex post equilibrium; for robust implementation, we must rule out bad Bayesian, or interim equilibria on all type spaces

20 Back to the Single Object Example... robust implementation fails even in the private values case, since truthtelling is only a weak best response and there are many equilibria leading to ine cient outcomes in second price sealed bid auctions. robust implementation of the e cient allocation is not possible in the single object example (with private or interdependent values) even if augmented (but well-behaved) mechanisms are allowed. but robust implementation is achievable for a nearly e cient allocation under additional restrictions...

21 ...and to Private Values: The Modi ed Second Price Auction with probability 1 ε allocate object to highest bidder and pay second highest bid for each i, with probability ε b i I i gets object and pays 1 2 b i truth-telling is a strictly dominant strategy and ε-e cient allocation is robustly implemented

22 Interdependent Values: Modi ed VCG Mechanism with probability 1 ε allocate object to highest bidder i and winner pays max j6=i for each i with probability fb j g + γ b j j6=i i gets object and pays ε b i I 1 2 b i + γ j6=i truth telling is a strict ex post equilibrium b j

23 The Modi ed Generalized VCG Mechanism but existence of strict ex post equilibrium does not imply robust implementation in fact, this mechanism robustly implements the e cient outcome if and only if jγj < 1 I 1 and no mechanism robustly implements the e cient outcome if jγj 1 I 1 c.f. Chung and Ely (2001)

24 Rationalizability: A Key Epistemic Result A message m i can be sent by an agent with payo type θ i in an interim equilibrium on some type space if and only if m i is "incomplete information rationalizable" in the following sense: 1 First, suppose that every payo type θ i could send any message m i 2 Delete those messages m i that are not a best response to some conjecture over payo type - message pairs of the opponents that have not yet been deleted 3 Repeat step 2 until you converge

25 Illustrate Rationalizability in Generalized VCG mechanism direct mechanism: message m i is report θ 0 i agent i conjectures that other agents have type θ i and report θ 0 i : λ i θ 0 i, θ i 2 (Θ i Θ i ) β k i (θ i ) is set of reports i might send for some conjecture λ i θ 0 i, θ i over his opponents types θ i and reports θ 0 i, with restriction on conjecture λ i θ 0 i, θ i that each type θj of agent j sends a message in β k 1 i (θ j ) initialize at step k = 0 by allowing all reports β 0 i (θ i ) = [0, 1]

26 Linear Valuation Model with linear interdependence: v i (θ) = θ i + γ θ j j6=i ex post incentive compatible transfer yi (θ) is quadratic in reports θ 0 agent i with type θ i has linear best response θ 0 i : θ 0 i = θ i + γ j6=i θ j θj 0 linear best response allows us characterize set of best response reports by β k i (θ i ): β k i (θ i ) = h β k i i (θ i ), β k i (θ i )

27 Rationalizability with linear best response θ 0 i : θ 0 i = θ i + γ j6=i θ j θj 0 the upper bound β k (θ i ) is determined by: β k (θ i ) = minf1, θ i + γ max f(θ 0 i,θ i):θj 0 2β k j 1 (θ j ), 8j6=ig (θ j θj 0 )g j6=i or through lower bound β k 1 j (θ j ): β k (θ i ) = minf1, θ i + γ max θ i (θ j β k 1 (θ j j ))g j6=i

28 Induction rewriting: β k (θ i ) = minf1, θ i + γ max θ i (θ j β k 1 (θ j j ))g j6=i we obtain and likewise n o β k (θ i ) = min 1, θ i + (γ (I 1)) k, n o β k (θ i ) = max 0, θ i (γ (I 1)) k. thus θ 0 i 6= θ i ) θ 0 i /2 β k (θ i ) for su ciently large k, provided that γ < 1 I 1

29 On the other hand... but now suppose that γ > 1 I 1 each type θ i convinced (believes with probability one) that others types are θ j = γ (I 1) 1 2 θ i now the expected value of the object for i is 1 θ i + γ (I 1) θ i = 1 [1 + γ (I 1)] γ (I 1) 2 2 types cannot be distinguished (and hence separated) in direct or any other mechanism...

30 In the single unit auction example: robust implementation possible (using the modi ed generalized VCG mechanism) if jγj < 1 I 1 robust implementation impossible (in any mechanism) if γ > 1 I 1

31 In general environment... each Θ i is a compact subset of the real line agent i s preferences depend on θ through h i : Θ! R preferences are single crossing in h i (θ) Theorem (2009) 1 Robust implementation is possible in the direct mechanism if strict EPIC and the "contraction property" hold. 2 Robust implementation is impossible in any mechanism if either strict EPIC or the "contraction property" fails.

32 Contraction Property "deception": β = (β 1,..., β I ); β i : Θ i! 2 Θ i? with θ i 2 β i (θ i ) "truth-telling": β = (β 1,..., β I ) with β i (θ i ) = θ i the aggregator functions h satisfy the strict contraction property if, 8β 6= β, 9i, θ 0 i 2 β i (θ i ) with θ 0 i 6= θ i, such that sign θ i θ 0 i = sign hi (θ i, θ i ) h i θ 0 i, θ 0 i, for all θ i and θ 0 i 2 β i (θ i ) in the linear model this is equivalent to jγj 1 I 1

33 Contraction Property 2 with linear aggregator for each i: h i (θ) = θ i + γ ij θ j j6=i the contraction property satis ed if and only if largest eigenvalue of the interaction matrix: jγ 12 j jγ 1I j Γ, jγ 21 j jγ I 1I j 5 jγ I 1 j jγ II 1 j 0 is less than 1.

34 The Role of the Common Prior what if we know that the common prior assumption holds? in the analysis so far, no restrictions were placed on agents beliefs and higher order beliefs consider the role of beliefs and hence intermediate notions of robustness remain in the linear valution model with linear best responses now not only the size but also the sign of the interdependence, γ, matters

35 Strategic Complements recall the linear best response in the auction model θ 0 i = θ i + γ j6=i θ j θj 0 negative interdependence in agents types, γ < 0, gives rise to strategic complementarities in the direct mechanism restricting attention to common prior type spaces makes no di erence, and the contraction property continues to play the same role as described earlier Milgrom and Roberts (1991): with strategic complementarities, there are multiple equilibria if and only if there are multiple rationalizable actions)

36 Strategic Substitutes recall the linear best response in the auction model θ 0 i = θ i + γ j6=i θ j θj 0 positive interdependence in agents types, γ > 0, gives rise to strategic substitutability in the direct mechanism, and robust implementation becomes easier in particular, it is often possible even if the contraction property failed: if 1 I 1 < γ < 1, robust implementation is possible if we restrict attention to type spaces satisfying the common prior assumption

37 Future Questions Local, Intermediate Notions of Robustness Robust Predictions for Revenue Maximization Problem Single Crossing Conditions in Rich Type Spaces Beyond Mechanism Design: Robust Predictions In Games With Private Information If we cannot make unique predictions, can we provide robust bounds on the distribution of outcomes.

38 add interdependent preferences add strength of interdependence as an argument add argument for uniqueness are di erent emphasize di erent values of γ emphasize di erent proof techniques have example of budget balancing as counterexample